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LearningLearning ObjectivesObjectives ..

-- Graphs &TransformationsGraphs &Transformations

The student will become familiar with a beginningelementary functionselementary functions.

and horizontal shiftsand horizontal shifts.

e s u en w e a e u uu ustretches and shrinksstretches and shrinks.

The student will be able to graph piecewiseto graph piecewisedefidefifunctionsfunctions.

Calculus for Business & Economics

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ProblemProblem

=

x y = x2

-3

--1

1

2


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SolutionSolution

=

x y = x2

-3 9

--1 1

1 1

2 4


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ProblemProblem

Sketch the graph of f(x) = (x 2)2

and explain, it s re ate to t e grap o x = x :

x y = (x-2)2

-3 25

--1 9

1 1

2 0


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Com arison o x = xCom arison o x = x22and x =and x =


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SolutionSolution continuedcontinued

difference is that the original graph has been tran

.

2shifted horizontally two units to the right on thshifted horizontally two units to the right on th

Notice that replacing x by xreplacing x by x 22shifts the graph

Correspondingly, replacing x by x + 2replacing x by x + 2 would sh


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ProblemProblem

= 3 = 3


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SolutionSolution

= 3get the graph of f(x)=x3 + 5.


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ProblemProblem

= = and find the domains.


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SolutionSolution

of the graphs about the x-axis, we get the graph o


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ProblemProblem

3= 3=


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SolutionSolution

3=to the left, we get the graph of 3 1f ( x ) x= +


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Absolute Value FunctionAbsolute Value Function a x x=


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Absolute Value FunctionAbsolute Value Function a( x) x =continuecontinue


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Summar o Gra h Trans ormationsSummar o Gra h Trans ormations

1.

Vertical

1.

Vertical

TranslationTranslation: = f x + k(1) k > 0: Shift graph of y = f (x) up k units.

(2) k < 0: Shift graph of y = f (x) down |k| units

2.

Horizontal

2.

Horizontal

TranslationTranslation: y = f (x+h)

(1) h > 0: Shift graph of y = f (x) left h units.

3.

Reflection3.

Reflection: y = f (x).

Reflect the ra h of = f x in the xaxis.

4.

Vertical

4.

Vertical

Stretch

and

ShrinkStretch

and

Shrink: y =Af(x)

(1) A > 1: Shrink graph of y = f (x) vertically by

each coordinate value by A.(2) 0 < A < 1: Stretch graph of y = f (x) vertical


.

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PiecewisePiecewise--De ined FunctionsDe ined Functions

be defined as x , if x 0

>

x , if x 0

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Exam leExam le

2 2x x 2 ,


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SolutionSolution

2 2x x 2 ,


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End lid End lid

, ,applications given in the textbook.

If you dont solve a problem by your hands, yo

.of understanding the concepts/techniqueunderstanding the concepts/technique

,,


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Learning ObjectivesLearning Objectivesfor Section 2.3 Solving Quadratic Fufor Section 2.3 Solving Quadratic Fu

The student will be able to identify and define qto identify and define qunctions e uations and ine ualitiesunctions e uations and ine ualities.

The student will be able to identify and use proto identify and use pro

uadratic unctions and their ra hsuadratic unctions and their ra hs.The student will be able to solve applications ofto solve applications of

unctions.unctions.The student will be able to graph and identify pto graph and identify p

ol nomial and rational unctionsol nomial and rational unctions.


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a 02f ( x ) ax bx c= + +

is a quadratic functionquadratic function and its graph is called a


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function2f ( x ) ax bx c= + +

to what is known as the vertex formvertex form:

2f ( x ) a( x h) k= +

,,


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Completing the SquareCompleting the SquareTo find the Vertex of a Quadratic FuTo find the Vertex of a Quadratic Fu

The example below illustrates the procedure:Consider 2

=


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2 .2 2f ( x ) 3x 6 x 1 3[ x 2x]= + =

Step 2. Recall the formula2 2x 2ax a ( x+ + = +

, 2 22x ( 1) ( 1f ( x ) 3[ x + =

2

x x

3 x 1 2

= + +

= +

Step 3. From the vertex form, we deduce the verte


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and( something , ) ( , some0 0 thing)

The first one is called thexxinteinte( something ,0)

of the function.

Example) has the xintercept

f ( x ) 2x 3= 3

(2


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2f ( x ) 3x 6 x 1= +


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=20 3x 6 x 1= +

By the Quadratic FormulaQuadratic Formula, we have

acx

2a 2( 3)

= =

6 240.18 , 1.82

=

Therefore, we have twoxxinterceptsintercepts: a( 0.18 ,0)


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2x 3x= +implies f ( 0 ) 3( 0) 6( 0) 1 1= + =

Therefore, we have only oneyyinterceptintercept: ( 0 , 1


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2x a x h k= +1. If , then the graph of f(x) is aparabolaparabola.a 0

(2) If , then the graph is concave downwconcave downwa 0 { y


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2


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2

SolutionSolution

This inequality holds for those values of x for whiof f x is at or above the xaxis. This happens fothe two x intercepts, including the intercepts. (If yt egrap o t e unction, you can un erstan teasily.) Thus, the solution set for the quadratic in

0.18 x 1.82


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ease, a e a oo a e a ema ca e ec o


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A lication o Quadratic FunctionsA lication o Quadratic Functions

, ,acreacre. Each tree produces, on the average, 300 pea300 pea

,peaches per tree is reduced by 10reduced by 10.

How many more treesHow many more trees should the farmer plant to

What is the maximummaximum yield?


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luti nluti n

=

Yield = (300)(20) = 6000 (currently)

an one more ree:Yield = (300 1(10))(20 + 1)

= = peac es.

Plant two more trees:Yield = (300 2(10))(20 + 2) = (280)(22)

Plantxmore trees:

Yield = (300 x(10))(20 +x)2


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luti n ntinu d luti n ntinu d

= + +

2Y( x ) 10x 100x 6000

quadratic function of which graph is concave dow

value of the yield. Graph is below, with the y val

To find the vertex, wecom lete the s uare.


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2Y( x ) 10( x 10x ) 6000= +

2

10 x 10x 5

10 x 5 10 25 60

5

00

= +

= + +

210( x 5 ) 6250= +

So, the farmer should plant 5 additional trees and

,

2

.function implies that the graph is concave downw


ver ex mus e e max mum.

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Br kBr kE nE n n l i n l i

cameras has the revenue and cost functionsfor x

R( x) x( 94.8 5 x)=

Both have domain

.=

1 x 15 Breakeven pointsare the production levels at w

R( x ) C( x )=

Find the break-even points algebraically to the nethousand cameras.


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=x( 94.8 5 x ) 156 19.7x = +

2

. x x . x

5 x 75.1x 156 0

= +

+ =

By the Quadratic FormulaQuadratic Formula, we get2

x2a

=

275.1 ( 75.1) 4( 5 )(156 ) 2.49

2 5

=


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x 2.49 x

If we graph the cost and revenue functions on a guti ity, we o tain t e o owing grap s, s owing tintersection points:


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u dr ti R r i nu dr ti R r i n

a parabola would be a better model of the data th

linear model to the data, we would use quadratic

2y( x ) ax bx c= + +

t at est its t e ata.

.


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P l n mi l Fun ti nP l n mi l Fun ti n

written in the form

n n 1 2=

The domain of a polynomial function is the set of

n n n 1 2

numbers.

.A polynomial of degree 1is a linear function.

.A polynomial of degree 3 is a cubic function.


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h P l n mi l h P l n mi l

if it only contains odd powers of x.

if it only contains even powers of x.

Lets look at the shapes of some even and odd poly

Symmetry Number of local maxima/minima -


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E m lE m l


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r ti n dd P l n mi l r ti n dd P l n mi l

For an odd ol nomial the graph is symmetric about the origin the graphs starts negative, ends positive, or vice ve

on whether the leading coefficient is positive or neg either way, a polynomial of degree ncrosses the x

, . For an even polynomial,

the ra h is s mmetric about the axis the graphs starts negative, ends negative, or starts

positive, depending on whether the leading coefficie

negative either way, a polynomial of degree ncrosses the x


. .

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h r t ri ti P l n mi l h r t ri ti P l n mi l

Gra hs o ol nomials are continuouscontinuous. One can skwithout lifting up the pencil.

Gra hs o ol nomials have no shar cornersno shar corners.Graphs of polynomials usually have turning pointturning point

point that separates an increasing portion of the grdecreasing portion.

A polynomial of degree ncan have at most nlineaere ore, t e grap o a po ynom a unct on o po

can intersect the xaxis at most ntimes.

times.


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R ti n l Fun ti nR ti n l Fun ti n

and Q(x), for all xsuch that Q(x) is not equal to z

Example: Let P(x) = x+ 5 and Q(x) = x 2,

xR(x)

x 2

+=

is a rational function whose domain is all real val


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V rti l m t t R ti n l FunV rti l m t t R ti n l Fun

verticalvertical asymptotesasymptotesto the graph of the function.

A vertical asymptote is a line of the form x = kw

figure below, which is the graph of

=

R xx 2

=


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H riz nt l m t t R ti n l FH riz nt l m t t R ti n l F

which the graph of the function approaches as xa

x 5+

For example, in the graph of

xx 2

=

= .


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Generalization about AsymptotesGeneralization about Asymptoteso a ona unc ons o a ona unc ons

The number of vertical asymptotes of a rationa= /d is at most e ual to the de ree

A rational unction has at most one horizontal

The ra h o a rational unction a roachesThe ra h o a rational unction a roacheshorizontal asymptote (when one exists) bothhorizontal asymptote (when one exists) bothincreases and decreases without bound.increases and decreases without bound.


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End lid End lid




,,


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ObjectivesObjectivesoror ection .ection . xponentiaxponentia unctionunction

The student will be able to graph and identify tto graph and identify tof exponential functionsof exponential functions.

The student will be able to apply base eexpone

functions, including growth and decay applicatgrowth and decay applicat The student will be able to solve compound inteto solve compound inte


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Ex onentialEx onential FunctionsFunctions

x =

the set of all positive real numbersall positive real numbers.


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RiddleRiddle

twotwo pennies on the second day of Decemberpennies on the second day of December, four four

, many pennies would you receive on December 31

sum payment of $10,000,000?


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SolutionSolution

Da No o Pennies Currentl To

Dec. 1 1=20 20

.

Dec. 3 4=22 20+21+22

Dec. 4 8=2 2 +2 +2 +2

Dec. n 2n-1 20+21+22+23++2n-

Dec. 31 230 20+21+22+23++2n-


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Solution continuedSolution continued

, ,have 230pennies on Dec. 31. The exponent on tw

.

Since 230

=10,737,418.24, which is bigger than $thus on Dec. 31, you should get $10,737,418.24.

s examp e s ows ow an exponen a unc oow an exponen a unc oextremely rapidlyextremely rapidly. In this case, the exponential f

is used to model this roblem.f ( x ) 2=


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Gra h oGra h o xx 2=

have the following graph:


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CharacteristicsCharacteristicsof the Graph of whenof the Graph of whenf ( x) b= b >

1. All graphs will approach the xaxis as x becomunbounded and negative.

2. All graphs will pass through (yyintercepintercep(0,1)3. There are no xno xinterceptsintercepts.4. Domain is all real numbers.5.5. Range is all positive real numbersRange is all positive real numbers.6. The graph is always increasing on its domain.

7. All graphs are continuous curves.


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CharacteristicsCharacteristicsof the Graph of whenof the Graph of whenf ( x) b= 0 x 3 3> xfor nega


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Relative Growth RatesRelative Growth Rates

ktkt ,and the independent variable t represents time, are

Note that if t = 0, then y = c. So, the constant cconstant cr

initial o ulation or initial amount.initial o ulation or initial amount. The constant kThe constant k is called the relative growth raterelative growth rate

rowth rate is k = 0.02, then at any time t, the pogrowing at a rate of 0.02y persons (2% of the popyear.

We say thatpopulation is growing continuouslypopulation is growing continuouslygrowth rate kgrowth rate kto mean that the population y is giv


modely t = cy t = c ee .

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Growth Deca A licationsGrowth Deca A lications-- Atmospheric PressureAtmospheric Pressure

The atmospheric pressure P decreases with increaThe pressure is related to the number of kilometersea level by the formula: P(h)=760e-0.145h

Question 1. Find the pressure at sea level (h=0).

Question 2. Find the pressure at height of 7 kilom


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SolutionSolution

= 0=

P(7)=760e-0.145(7)=275.43


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De reciation o a MachineDe reciation o a Machine

value each year. Its value after t years is given by

= t .

$30,000.

Solution) V(8)=30000(0.98)=$12,914


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Com ound InterestCom ound Interest

nt

rA P 1 = +

n

amount (principal), r is the annual interest rate a

is the number of years.


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Exam leExam le

bank at 5.75% interest compounded quarterly for


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SolutionSolution

bank at 5.75% interest compounded quarterly for

Solution) Use the compound interest formula:

rA P 1n

= +

Putting P=1500, r=0.0575, n=4, t=5, we obtai4 ( 5 )

$.A 1500 1 1994= + =


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End lid End lid




,,


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LearningLearning ObjectivesObjectivesoror Section 2.5Section 2.5 Logarit micLogarit mic FunctionFunction

The student will be able to use and apply inverto use and apply inver The student will be able to use and a l lo arto use and a l lo ar

functions and properties of logarithmic functionfunctions and properties of logarithmic function

The student will be able to evaluate lo arithmsto evaluate lo arithms


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Lo arithmic FunctionsLo arithmic Functions

an exponential function inverse relationinverse relation.

We will study the concept of inverse functionsinverse functionsas


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OneOnetotoOne FunctionsOne Functions

. is necessary to discuss the topic of onetoone fu

, .

inputsinputsof a function correspond to distinct outpudistinct outpu

,

1 2 1 2For x x , f ( x ) f ( x )


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Gra h o OneGra h o OnetotoOne FunctionsOne Functions

of the one-to-one function should be different.

Lots of graphs will be given in class.


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Horizontal Line TestHorizontal Line Test

,pass the vertical line testpass the vertical line test. That is, a vertical line t

the graph only once at each x value.

There is a similar geometric test to determine if a - -

horizontal line drawn through the graph of a one-

crosses a graph more than once, then the function- -


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De inition o Inverse FunctionDe inition o Inverse Function

-- -- ,interchanging the x and y values of the original fuinterchanging the x and y values of the original fu

, ,function, then the ordered pair (b, a) belongs to th

.

-- --test), then the inverse of such a function does not inverse of such a function does not


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Lo arithmic FunctionsLo arithmic Functions

inverse of the exponential function with base 2inverse of the exponential function with base 2

Notice that the exponential function y=2x is one-

>to be the inverse of the exponential function with inverse of the exponential function with


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Inverse o an Ex onential FunctionInverse o an Ex onential Function

= xNow, interchange x and y coordinates : x=2y

There are no algebraic techniquesno algebraic techniques that can be use

,with base 2with base 2.

The definition of this new function is:

2og x y an on y x = =


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Graph, Domain, RangeGraph, Domain, Rangeo Logarit mic Functionso Logarit mic Functions

The domain of the logarithmic function y = log2xas the ran e o the ex onential unction = 2x. W

The ran e o the lo arithmic unction is the same domain of the exponential function (Again, why?)

Another fact: If one graphs any one-to-one functinverse on the same rid, the two ra hs will alwasymmetric with respect to the line y = x.


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Lo arithmicLo arithmicEx onential ConversionEx onential Conversion

logarithmic into an exponential expression andlogarithmic into an exponential expression and

4log 16 x 16 4 x 2= = =

33 3 33

log log log 3 327 3

= = =

81281 log 81 9 9 9

2= = =

35log125 5 125 3= =


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Solvin E uationsSolvin E uations

,involving logarithms.

Examples: 33b b b 1log 1000 3 10 ===

56log x 5 6 x 7776 x = = =

In each of the above, we converted from log form


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Pro erties o Lo arithms Must MemPro erties o Lo arithms Must Mem

real numbers, b not equal to 1, and p and x are re

xb b b1. log 1 0 2. log b 1 3. log b x= = =

b b b5. log N

M

MN) log M log = +

b b b

p

.N

=

=

b b

.

8. log M log N if and only if M N = =


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Exam leExam le

=


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SolutionSolution

=

4log ( x 6 )( x 6 ) 3+ =

4

3 2

log ( x 36 ) 3 =

= 264 x 36 =

2100 x

10 x

=

=

x 10=


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Exam leExam le

=


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SolutionSolution

=

10log x

=

1

lo x

=4

10

1000

l

0

10og x =

x 4=


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Common Lo s Natural Lo sCommon Lo s Natural Lo s

=If no base is indicated, the logarithm is assumed t

Natural logNatural log:

eln x log x =


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Exam leExam le

=


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SolutionSolution

=

x 1ln 1

+=

xx 1

e

+

= xex x 1= +

( e 1)x 1 =

x e 1=


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A licationA lication

at 4 % interest?


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SolutionSolution

nt

rA P 1 = +

12t

n

0.04

12t

12

=

( ) ( 12t

.

ln2 ln 1.00333 12t ln 1.003= =

ln2 t t 17.36 12 ln 1.00333

==


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Lo arithmic Re ressionLo arithmic Re ression

,bases increase much more slowly for large vb 1>

.inspection of the plot of a data set indicates a slow

,

the function of the form that best fiy na xb l= +

Again, since the regression subject depends on the


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End lid End lid




,,


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Chapter 2 ReviewChapter 2 Review-- mportant erms, ym o s, oncepmportant erms, ym os, oncep

2.1. Functions2.1. FunctionsPointPoint--byby--point plottingpoint plottingmay be used to sketch theequation in two variables: plot enough points from

set in a rectangular coordinate system so that theapparent and then connect these points with a sm

Afunctionfunction is a correspondence between two sets osuch that to each element in the first set there cor

and only one element in the second set. The first sthe domaindomain and the second set is called the rangerange.


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Section 2.1 Functions continuedSection 2.1 Functions continued

is the independent variable or input. If y represen

output.

If in an equation in two variables we get exactly o

such a function is just the graph of the equation. I

specify a function.


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an equation in two variables specifies a function.

The functions specified by equations of the formyy

Functions s eci ied b e uations o the orm = b= bconstant functionsconstant functions.


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indicated, we agree to assume that the domain is

corresponds to the element x of the domain.

BreakBreak--eveneven and profitprofit--loss analysisloss analysis uses a cost fu

loss (Rloss (R >


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Section 2.2 Elementary FunctionsSection 2.2 Elementary Functions Grap s & Trans ormationsGrap s & Trans ormations

The six basic elementary functions are the identitidentitthe s uare and cube unctions the s uare root anthe s uare and cube unctions the s uare root anfunctions and the absolute value functionfunctions and the absolute value function.

transformation of the graph of the function.

e as c grap rans orma ons are: ver ca anver ca antranslations (shifts), reflection in the xtranslations (shifts), reflection in the x--axis, and axis, and

s re c es an s r n s sre c es an s r n s .A piecewise-defined function is a function whose


involves more than one formula.

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Section 2.3Section 2.3 uadraticuadratic FunctionsFunctions

function f(x) = ax2 + bx + c is a quadratic funcquadratic func

22b b 4ac x when b 4ac

=

-

2a


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Section 2.3Section 2.3 uadraticuadratic Functions conFunctions con

function produces the vertex form,vertex form,

--From the vertex form of a quadratic function, we

, ,, , ,,minimum, and range, and sketch the graphminimum, and range, and sketch the graph.

If a revenue function R(x) and a cost function C(x

0, 0,referred to as breakbreak--even pointseven points.


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Section 2.3Section 2.3 uadraticuadratic Functions conFunctions con

function of the form y = ax2 + bx + c that best f

A quadratic function is a special case of a polyno

f(x) = anxn + an-1x

n-1 + + a1x

Unlike polynomial functions, a rational function a rational function vertical as m totesvertical as m totes but not more than the de redenominator) and at most one horizontal asympat most one horizontal asymp


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Section 2.4 Ex onential FunctionsSection 2.4 Ex onential Functions

f (x) = bx, where b is not equal to 1, but is a posit

numbers and the rangerange is the set of positive real

The graph of an exponential functionThe graph of an exponential function is continu-

Ex onential unctionsobe the amiliar laws oobe the amiliar laws oand satisfy additional properties.


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Section 2.4 Ex onential Functions cSection 2.4 Ex onential Functions c

irrational number eirrational number e 2.71832.7183.

xponen a unc ons can e usegrowth and radioactive decaygrowth and radioactive decay.

Exponential regression on a graphing calculator pfunction of the form y = a(bx) that best fits a data

Exponential functions are used in computations ofnt

A P 1 n= +


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Section 2.5 Lo arithmic FunctionsSection 2.5 Lo arithmic Functions

-- --corresponds to exactly one domain value.

The inverse of a oneThe inverse of a one--toto--one functionone functionf is

variables of f. That is, (a, b) is a point on the grap,

-- --


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Section 2.5 Lo arithmic Functions cSection 2.5 Lo arithmic Functions c

the logarithmic functionlogarithmic function with base b, denoted y =

The domaindomain of logbx is the set of all positive real

.the inverse of the function y = bx,y =y = loglogbbx isx is ee== yy

corresponding properties of exponential functions


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Section 2.5 Lo arithmic Functions cSection 2.5 Lo arithmic Functions c

denoted by log xlog x. LogarithmsLogarithms withwith base ebase e are ca

the length of time it takes for the value of an inve

function of the form y = a + b ln x that best fits a


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End lid End lid




,,


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LearningObjectivesLearningObjectives

forSection3.1IntroductiontoLimforSection3.1IntroductiontoLim

The student will learn about:

unc ons an grap sunc ons an grap s

Limits from a graphic approachLimits from a graphic approach

Limits from an algebraic approachLimits from an algebraic approach Limits of difference uotients.Limits of difference uotients.


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Functions&GraphsFunctions&Graphs

BriefOverviewBriefOverview

The graph of a functiongraph of a function is the graph of

or ere pa rs a sa s y e unc on.

Example: f ( x ) 2 x 1

=


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Analyzing

a

Limit

(Graphical

ApprAnalyzing

a

Limit

(Graphical

Appr

, as x goes to 3, f(x) = 2x 1 goes to 5

In fact, without using the graph, we obse

as x goes to 1000, f(x) = 2x 1 goes to

We introduce a notation and express as

,


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LimitLimit

,as x c, f(x) L is compacted into

clim f ( x ) L =

It reads as x goes to c, the limit of f(xas x goes to c, the limit of f(x

The meaning of the equation is whenev

, ,

e single real number L.


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OneOneSided

LimitSided

Limit

and call K the limit from the leftlimit from the left (or leftleft

x c =

if f (x) is close to K whenever x is close

e left of c on the real number line.

We write x clim f ( x ) L+ =

mitmit) if f (x) is close to L whenever x is cl

o the right of c on the real number line.

,

the limit from the right must exist and be


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Example

Example

((cconfertheonfertheMathematicaMathematica

x 2lim

x 2ut m

x 2,

xlim

xan m

x ,


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Limit

Properties

(Algebraic

ApproLimit

Properties

(Algebraic

Appro

,following two limits exist and are finite

x c x clim f ( x ) L and lim g( x =

Then (1) the limit of the sumlimit of the sum of the fuual to the sum of the limitssum of the limits and the difference of the functions is equal to

x clim [ f ( x ) g( x )] + x clim [ f ( x )

x c x clim f ( x ) lim g( x ) = + x c lim f ( x ) =

= Calculus for Business & Economics

=

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Limit

Properties

(Algebraic

ApproLimit

Properties

(Algebraic

Appro

al to the constant times the limitconstant times the limit of t

: x c x clim [ af ( x )] a[lim f ( x )] aL , a = =

(3) The limit of the productlimit of the product of the funroduct of the limitsroduct of the limits of the functions

x c x c x clim [ f ( x ) g( x )] [lim f ( x )][lim g =


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Limit

Properties

(Algebraic

ApproLimit

Properties

(Algebraic

Appro

quotient of the limitquotient of the limit of the function, s no equa o :

x lim x Lx c x

x c

m , m

g( x ) lim g( x ) K

= =

(5) The limit of the nlimit of the n--thth rootroot of a fun--

1 1

n n = x c x c


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ExampleExample

2 2

x 2 x 2 x 2

lim 2x

x 4

x 4lim

lim( 3 x 13x 1 13)

=

+ =

+

From these examples, we conclude tha ,

lim ( x ) ( c )=

(2) for any rational function R(x) with a

c

nominator at x = c,

=Calculus for Business & Economics

c

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Indeterminate

FormsIndeterminate

Forms

c x clim f ( x ) 0 and lim g( x ) 0 = =

en, f ( x ) g( x )orlim lim

is said to be indeterminateindeterminate. The term te is used because the limit may or m

How to make the indeterminate formHow to make the indeterminate form


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ExamplesExamples

2 x 2 x 2 x 2lim lim lim

x 2 x 2 = =

2x 1 x 1 x

1)lim lim lim

( x ( x 1)

x 1 x 1 x 1

= = +


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Difference

QuotientsDifference

Quotients

= , h 0lim h


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SolutionSolution

= , h 0lim h

Solution: f ( a h) 3( a h) 1 3a 3+ = + = +

f ( a) 3a 1=

f ( a h) f ( a) 3h

=

+ h 0 h 0h h

= =


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SummarySummary

dea of a limitdea of a limit. This was an intuitive wac m ts.

nt were the same, we had a limit at nt were the same, we had a limit at We saw that we could add, subtracadd, subtracand divide limitsand divide limits.

We now have some very powerful tng w m s an can go on o our sus.


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3.7 Marginal Analysis in Business and Economics

Definition.

marginal cost function

marginal revenue function

marginal profit function

Remark.

Example.

Theorem.

Example.

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4.1 The Constant and Continuous Compound Interest

Theorem .

Example.

Example.

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4.2 Derivatives of Exponential and Logarithmic Functions

Theorem .

Example.

Theorem .

Example.

Theorem .

Example.

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4.3 Derivatives of Products and Quotients

Theorem .

Example.

Theorem .

Example.

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4.4 The Chain Rule

Definition. composite

Example.

Exercise.

Theorem .

Example.

Exercise.

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4.5 Implicit Differentiation

explicitly

implicitly

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Example.

Exercise.

Exercise.

Exercise.

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5.1 First Derivative and Graphs

Increasing and Decreasing Functions

increasing

decreasing

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Theorem 1 .

Discussion 2.

4 2 2 4x

10

20

fxx2

4 2 2 4x

2

4

gxx

Example 3.

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1 0 1 3 5 7x

1

5

10

17

fxx26x10

Definition 4 .

critical values

Example 5.

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1 1x

1

fxx3

Example 6.

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5 3 1 1 3x

1

1

fx1x13

Example 7.

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5 3 1 1 3 5x

1

1

fx1

x

Example 8.

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1 5 10x

3

fx5Logxx

Discussion 9.

Exercise 10.

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Local Extrema

Definition 11. local maximum

local minimum

local extremum

turning point

Example 12.

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Theorem 13 .

FirstDerivative Test

Theorem 14 .

positive rising/increasing

negative falling/decreasing

local maximum

negative falling/decreasing

positive rising/increasing

local minimum

Example 15.

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1 2 3 4 5 6

6

10

fxx39x

224x10

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Applications to Economics

6 16 20t

4

1

s't

Example 16.

FirstDerivative

Test


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6 16 20t

St

Example 17.

100 150 200 250x

1000

125

Px

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100 150 200 250x

P'x

ln.pdf

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